Binary Operations

1
Binary Operations
Math/Logical
2
Binary Math
3
Decimal Addition Example
1) Add 8 7 15 Write down 5, carry 1
Add 3758 to 4657
2) Add 5 5 1 11 Write down 1, carry 1

• 3 7 5 8
• 4 6 5 7

1
1
1
3) Add 7 6 1 14 Write down 4, carry 1
8
4
1
5
4) Add 3 4 1 8 Write down 8
4
Decimal Addition Explanation

• What just happened?
• 1 1 1 (carry)
• 3 7 5 8
• 4 6 5 7
• 8 14 11 15 (sum)
• 10 10 10 (subtract the base)
• 8 4 1 5

• 1 1 1
• 3 7 5 8
• 4 6 5 7
• 8 4 1 5

So when the sum of a column is equal to or
greater than the base, we subtract the base from
the sum, record the difference, and carry one to
the next column to the left.
5
Binary Addition Rules

• Rules
• 0 0 0
• 0 1 1
• 1 0 1 (just like in decimal)

• 1 1 210 102 0 with 1 to carry
• 1 1 1 310 112 1 with 1 to carry

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Binary Addition Example 1
Col 1) Add 1 0 1 Write 1
Example 1 Add binary 110111 to 11100
Col 2) Add 1 0 1 Write 1
Col 3) Add 1 1 2 (10 in binary) Write 0,
carry 1

• 1 1 0 1 1 1
• 0 1 1 1 0 0

1
1
1
1
Col 4) Add 1 0 1 2 Write 0, carry 1
Col 5) Add 1 1 1 3 (11 in binary)
Write 1, carry 1
1
0
1
0
0
1
1
Col 6) Add 1 1 0 2 Write 0, carry 1
Col 7) Bring down the carried 1 Write 1
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Binary Addition Explanation
In the first two columns, there were no carries.
What is actually happened when we carried in
binary?
In column 3, we add 1 1 2 Since 2 is equal
to the base, subtract the base from the sum and
carry 1.

• 1 1 0 1 1 1
• 0 1 1 1 0 0
• - .

1
1
1
1
In column 4, we also subtract the base from the
sum and carry 1.
In column 5, we also subtract the base from the
sum and carry 1.
2
2
2
3
In column 6, we also subtract the base from the
sum and carry 1.
2
2
2
2
In column 7, we just bring down the carried 1
1
0
1
0
0
1
1
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Binary Addition Verification
Verification 1101112 ? 5510 0111002
2810 8310 64 32 16 8 4 2
1 1 0 1 0 0 1 1 64 16 2
1 8310
You can always check your answer by converting
the figures to decimal, doing the addition, and
comparing the answers.

• 1 1 0 1 1 1
• 0 1 1 1 0 0

1
0
1
0
0
1
1
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Binary Addition Example 2
Verification 1110102 ? 5810 0011112
1510 7310 64 32 16 8 4 2
1 1 0 0 1 0 0 1 64 8 1
7310
Example 2 Add 1111 to 111010.

• 1 1 1 0 1 0
• 0 0 1 1 1 1

1
1
1
1
1
1
0
0
1
0
1
0
10
Decimal Subtraction Example

• Try to subtract 5 7 ? cant.
• Must borrow 10 from next column.

Subtract 4657 from 8025
Add the borrowed 10 to the original 5. Then
subtract 15 7 8.

• Try to subtract 1 5 ? cant.
• Must borrow 10 from next column.
• But next column is 0, so must go to
• column after next to borrow.

• 8 0 2 5
• - 4 6 5 7

1
7
9
1
1
1
Add the borrowed 10 to the original 0. Now
you can borrow 10 from this column.
8
6
3
3
Add the borrowed 10 to the original 1.. Then
subract 11 5 6
3) Subtract 9 6 3
4) Subtract 7 4 3
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Decimal Subtraction Explanation

• 8 0 2 5
• - 4 6 5 7

8
6
3
3

• So when you cannot subtract, you borrow from the
  column to the left.
• The amount borrowed is 1 base unit, which in
  decimal is 10.
• The 10 is added to the original column value,
  so you will be able to subtract.

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Binary Subtraction Explanation

• In binary, the base unit is 2
• So when you cannot subtract, you borrow from the
  column to the left.
• The amount borrowed is 2.
• The 2 is added to the original column value, so
  you will be able to subtract.

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Binary Subtraction Example 1
Col 1) Subtract 1 0 1
Col 2) Subtract 1 0 1
Example 1 Subtract binary 11100 from 110011
Col 3) Try to subtract 0 1 ? cant. Must
borrow 2 from next column. But next column
is 0, so must go to column after next to
borrow.
1
2
Add the borrowed 2 to the 0 on the right.
Now you can borrow from this
column (leaving 1 remaining).
0
0
2
2

• 1 1 0 0 1 1
• - 1 1 1 0 0

Add the borrowed 2 to the original
0. Then subtract 2 1 1
Col 4) Subtract 1 1 0
Col 5) Try to subtract 0 1 ? cant. Must
borrow from next column.
1
1
0
1
1
Add the borrowed 2 to the remaining 0. Then
subtract 2 1 1
Col 6) Remaining leading 0 can be ignored.
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Binary Subtraction Verification
Verification 1100112 ? 5110 -
111002 - 2810 2310 64 32 16
8 4 2 1 1 0 1 1 1
16 4 2 1 2310
Subtract binary 11100 from 110011
1
2
0
0
2
2

• 1 1 0 0 1 1
• - 1 1 1 0 0

1
1
0
1
1
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Binary SubtractionExample 2
Verification 1010012 ? 4110 -
101002 - 2010 2110 64 32 16
8 4 2 1 1 0 1 0 1
16 4 1 2110
Example 2 Subtract binary 10100 from 101001
2
0
0
2

• 1 0 1 0 0 1
• - 1 0 1 0 0

1
1
0
1
0
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Logical, Shift, and Rotate Operations
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Logical, Shift and Rotate Operations

• A particular bit, or set of bits, within the
  byte can be set to 1 or 0, depending on
  conditions encountered during the execution of a
  program.
• When so used, these bits are often called
  "flags".
• Frequently, the programmer must manipulate these
  individual bits - an activity sometimes known as
  "bit twiddling".
• The logical, shift, and rotate operations
  provide the means for manipulating the bits.

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Logical OR Rules

• OR Operations
• OR Results in 1 if either or both of the operands
  are 1.
•  
• OR Table
• 0 OR 0 0
• 0 OR 1 1
• 1 OR 0 1
• 1 OR 1 1

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Logical OR Operation

• To perform the OR operation, take one column at
  a time and perform the OR operation using the OR
  table.
• Ex 1 1 0 0 1 0 0 1 1
• OR 0 0 0 0 1 1 1 1
• 1 0 0 1 1 1 1 1

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Logical OR Examples
Ex 2 1 1 0 0 1 0 0 1 OR 0 0 0 0 1 0
1 0 1 1 0 0 1 0 1 1

• Ex 3 0 1 1 1
• OR 0 0 1 0 0 1 1 1

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Logical XOR Rules

• XOR Operations
• The exclusive OR. Similar to OR except that it
  now gives 0 when both its operands are 1.
• Rules.
• 0 XOR 0 0
• 0 XOR 1 1
• 1 XOR 0 1
• 1 XOR 1 0

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Logical XOR Examples

• Ex 1 1 0 0 1 1 0 0 1
• XOR 0 0 0 0 1 1 1 1
• 1 0 0 1 0 1 1 0

Ex 2 0 1 1 1 XOR 0 0 1 0
0 1 0 1
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Logical AND Rules

• AND Operations
• AND yields 1 only if both its operands are 1.
• Rules.
• 0 AND 0 0
• 0 AND 1 0
• 1 AND 0 0
• 1 AND 1 1
•  

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Logical AND Examples

•  Ex 1 1 1 0 1 0 0 1 1
• AND 0 0 0 0 1 1 1 1
  0 0 0 0 0 0 1 1

•  Ex 2 0 1 1 1
• AND 1 0 0 1
• 0 0 0 1

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Logical NOT

• NOT Operations
• NOT is a separate operator for flipping the bits.
• Rules.
• NOT 0 1
• NOT 1 0  

Example. NOT 1 0 1 0 0 1 0 1
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Shift and Rotate operations

• Whereas logical operations allow the changing of
  bit values in place,
• SHIFT and ROTATE operations allow bits to be
  moved left or right without changing their
  values.

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Shift Left operation

• SHL
• SHL (shift left) shifts each bit one place to
  the left. The original leftmost bit is lost and
  a 0 is shifted into the rightmost position.
• Ex 1. SHL 1 1 0 1

0
1 1 0 1
1 0 1 0

• Ex 2. SHL 1 1 0 0
• 1 0 0 0

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Shift Left - Multiple Bits

• SHL bits means to shift left times
• Ex 1 SHL 3 1 0 0 1 1 1 0 0

1 0 0 1 1 1 0 0
0 0 0
1 1 1 0 0 0 0 0
Ex 2 SHL 2 1 1 1 0 0 1 1 0 1 0 0
1 1 0 0 0
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Shift Right operation

• SHR
• SHR (shift right) shifts each bit one place to
  the right. The original rightmost bit is lost
  and a 0 is shifted into the leftmost position.
• Ex 1. SHR 1 0 1 1

0 1 0 1 1
0 1 0 1
Ex 2. SHR 0 0 1 1 0 0 0 1
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Shift Right Multiple Bits

• SHR bits means to shift right times
• Ex 1 SHR 3 1 0 0 1 1 1 0 0
• 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 0 1 1

Ex 2 SHR 2 1 1 1 0 0 1 1 0 0 0
1 1 1 0 0 1
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Arithmetic Shift Right operation

• ASR (retains rightmost sign bit)
• Shifts each bit one place to the right. The
  original rightmost bit is lost and a the value of
  the most significant bit (leftmost bit) is
  shifted into the new leftmost position.
• Ex 1. ASR 1 0 1 1

Ex 2. ASR 0 0 1 1 0 0 0 1
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Arithmetic Shift Right Multiple Bits

• ASR bits means to arithmetic shift right
  times
• Ex 1 ASR 3 1 0 0 1 1 1 0 0
• 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1

Ex 2 ASR 2 0 1 1 0 0 1 1 0 0 0
0 1 1 0 0 1
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Rotate Left operation

•  ROL
• ROL (rotate left) shifts each bit one place to
  the left. The original leftmost bit is shifted
  into the rightmost position. No bits are lost.
• Ex 1. ROL 1 1 0 1

1 0 1
1
 Ex 2. ROL 1 1 0 0 1 0 0 1
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Rotate Left Multiple Bits

• ROL bits means to rotate left times
• Ex 1 ROL 3 1 0 0 1 1 1 0 0
• 1 1 1 0 0 1 0 0

Ex 2 ROL 2 1 1 1 0 0 1 1 0 1 0 0
1 1 0 1 1
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Rotate Right operation

•  ROR
• ROR (rotate right) shifts each bit one place to
  the right. The original rightmost bit is shifted
  into the leftmost position. No bits are lost.
• Ex 1. ROR 1 0 1 1

1 0 1
1

•   Ex 2. ROR 0 0 1 1
• 1 0 0 1

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Rotate Right Multiple Bits

• ROR bits means to rotate right times
• Ex 1 ROR 3 1 0 0 1 1 1 0 0
• 1 0 0 1 0 0 1 1

Ex 2 ROR 2 1 1 1 0 0 1 1 0 1 0 1
1 1 0 0 1